Is the Pope’s Authority Dependent Upon the Church?

You equate the Pope speaking “on behalf of the whole Church” with his speaking, as Vatican I states, “in the exercise of his office as shepherd and teacher of all Christians”. Your argument seems to turn on this equation. However, this would imply that Papal infallibility is purely a function of the infallibility of the Church, meaning that, as the Orthodox hold, the ultimate authority is the Church as a whole. However, this conclusion is directly antithetical to the following, from the First Vatican Council: “… such definitions of the Roman Pontiff are of themselves, and not by the consent of the Church, irreformable,” making the Pope, not the Church as a whole, the final and ultimate authority.

I agree in part. It’s true that I equate the Pope speaking “on behalf of the whole Church” with his speaking, as Vatican I states, “in the exercise of his office as shepherd and teacher of all Christians”. But it’s not true that Papal infallibility is purely a function of the infallibility of the Church. In Matthew 16:15, Jesus addresses the question, “Who do you say that I Am?” to the entire Twelve. Peter speaks up, inspired by God, and answers on their behalf. He doesn’t have to wait until the others had consented, because his power is coming from God, not from the consensus of the Twelve.

That said, when they do reach a final answer by consensus (like in Acts 15), it’s infallible as well under virtually the same conditions as a papal declaration, with one exception: a Church Council has to be approved by the Pope:

A council in opposition to the pope is not representative of the whole Church, for it neither represents the pope who opposes it, nor the absent bishops, who cannot act beyond the limits of their dioceses except through the pope. A council not only acting independently of the Vicar of Christ, but sitting in judgment over him, is unthinkable in the constitution of the Church; in fact, such assemblies have only taken place in times of great constitutional disturbances, when either there was no pope or the rightful pope was indistinguishable from antipopes.

This, at first brush, sounds like we’re declaring the Pope, not the Church as a whole, the final and ultimate authority. The Catholic Encyclopedia does a better job than I can explaining the nuances (same source as above):

Papal and conciliar infallibility are correlated but not identical. A council’sdecrees approved by the pope are infallible by reason of that approbation, because the pope is infallible also extra concilium, without the support of a council. The infallibility proper to the pope is not, however, the only formal adequate ground of the council’s infallibility. The Divine constitution of the Church and the promises of Divine assistance made by her Founder, guarantee her inerrancy, in matters pertaining to faith and morals, independently of the pope’s infallibility: a fallible pope supporting, and supported by, a council, would still pronounce infallible decisions. This accounts for the fact that, before the Vatican decree concerning the supreme pontiff’s ex-cathedra judgments, Ecumenical councils were generally held to be infallible even by those who denied the papal infallibility; it also explains the concessions largely made to the opponents of the papal privilege that it is not necessarily implied in the infallibility of councils, and the claims that it can be proved separately and independently on its proper merits. The infallibility of the council is intrinsic, i.e. springs from its nature. Christ promised to be in the midst of two or three of His disciples gathered together in His name; now an Ecumenical council is, in fact or in law, a gathering of all Christ’s co-workers for the salvation of man through true faith and holy conduct; He is therefore in their midst, fulfilling His promises and leading them into the truth for which they are striving. His presence, by cementing the unity of the assembly into one body — His own mystical body — gives it the necessary completeness, and makes up for any defect possibly arising from the physical absence of a certain number of bishops. The same presence strengthens the action of the pope, so that, as mouthpiece of the council, he can say in truth, “it has seemed good to the Holy Ghost and to us”, and consequently can, and does, put the seal of infallibility on the conciliar decree irrespective of his own personal infallibility.

Some important consequences flow from these principles. Conciliar decrees approved by the pope have a double guarantee of infallibility: their own and that of the infallible pope. The council’s dignity is, therefore, not diminished, but increased, by the definition of papal infallibility, nor does that definition imply a “circular demonstration” by which the council would make the pope infallible and the pope would render the same service to the council.

Here’s why papal consent is important: various episcopal groupings have come together and purported to be Church Councils. Since there are no clear rules regarding how many bishops are required (and since we’re never talking about 100% of the bishops), there needs to be some way to distinguish authentic Councils from inauthentic ones, and there have been inauthentic ones, just as there have been antipopes. Papal approval does this, since we know a valid pope can speak for the Church. Conversely, we can tell a valid pope from an invalid one by the valid mechanism (presently the College of Cardinals). So just as the pope confirms which conciliar groupings are valid, conciliar groupings confirm which men are real popes.

So papal consent is simply verification that this Council is speaking with the voice of the Church: they’re mutually supporting. This model is pretty consistent with what we see Above: the Father introduces the Son, the Son reveals the Father and introduces the Holy Spirit, the Holy Spirit reveals both the Father and the Son, etc. Neither of the three “veto” or overrule the other two, but the Holy Spirit can help dispel false apparations of Christ, and the witness of Christ helps to prevent us from false movements of the Spirit. So each part of the Trinity is valuable for safeguarding against false manifestations of the other Two, while none override the other Two.

The earthly trinity of living infallible bodies (infallible papal declarations, infallible Church Councils, and infallible sensum fidelis) operate the same way. We can tell is something is an infallible papal declaration if it comes from someone the Church has declared the pope (he doesn’t declare himself), and if it rises to the standards set out by Vatican I; we can tell authentic sensum fidelis by how it comports with known papal and conciliar de fide statements, etc. So I think it’s inaccurate to say that the pope is “the final and ultimate authority.” It’s more accurate to say that there are three interconnected and mutually supporting ways in which the Holy Spirit operates to ensure that Scripture and Tradition remain protected.